Finite Group Databasebeta

← C53×#12800 #27220312

Subgroups — C53×#12800 #27220312

Order 13568.

524 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2326C211yesmaximal, characteristic
2C2×C53×#354C212yesmaximal
3C53×#1799C212yesmaximal
4C53×#1393C212yesmaximal
5C2×#124×C53C2211yescharacteristic
6C2×C4×Q8×C53C411yescharacteristic, complemented
7C2×C4×D4×C53C2211yescharacteristic
8C2×#99×C53C2214yes—
9C53×#391C411yescharacteristic, complemented
10C53×#390C2211yescharacteristic
11C53×#388—22no—
12C53×#385—22no—
13C53×#384C411yescharacteristic, complemented
14C53×#380—22no—
15C53×#376C411yescharacteristic, complemented
16C53×#354—24no—
17C53×#342—22no—
18C22×Q8×C53C2×C411yescharacteristic
19C22×D4×C53C2311yescharacteristic
20C23×C4×C53D412yes—
21#124×C53—22no—
22#124×C53D412yes—
23#124×C53Q812yes—
24C4×Q8×C53—42no—
25C4×D4×C53—42no—
26C2×C4⋊C4×C53—22no—
27C2×C4⋊C4×C53—21no—
28C2×C4⋊C4×C53—42no—
29C2×C4⋊C4×C53C2×C411yescharacteristic
30C2×C4⋊C4×C53C2×C411yescharacteristic
31C2×C22⋊C4×C53—22no—
32C2×C22⋊C4×C53—22no—
33C2×C22⋊C4×C53—24no—
34C2×C22⋊C4×C53D412yes—
35C2×C42×C53—21no—
36C2×C42×C53C2×C411yescharacteristic
37C2×C42×C53C2×C411yescharacteristic
38C2×C42×C53C2×C411yescharacteristic
39#99×C53—24no—
40#99×C53—24no—
41#95×C53—21no—
42#95×C53—22no—
43#95×C53—21no—
44#95×C53—42no—
45C24×C53C2×D412yes—
46C2×Q8×C53—21no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53C4⋊C412yescomplemented
50C2×D4×C53—41no—
51C2×D4×C53—41no—
52C2×D4×C53C2×D411yescharacteristic
53C2×D4×C53C2×Q811yescharacteristic
54C2×D4×C53#4112yes—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—44no—
62C22×C4×C53—82no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C4⋊C4×C53—41no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—82no—
71C4⋊C4×C53—84no—
72C22⋊C4×C53—24no—
73C22⋊C4×C53—24no—
74C22⋊C4×C53—42no—
75C22⋊C4×C53—44no—
76C22⋊C4×C53—44no—
77C22⋊C4×C53—82no—
78C42×C53—22no—
79C42×C53—21no—
80C42×C53—42no—
81C42×C53—41no—
82C42×C53C4⋊C412yescomplemented
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—42no—
86C23×C53—42no—
87C23×C53C2×C22⋊C411yescharacteristic
88C23×C53#12112yes—
89C23×C53C22⋊Q812yes—
90Q8×C53—42no—
91Q8×C53—42no—
92D4×C53—41no—
93D4×C53—41no—
94D4×C53—42no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—41no—
102C2×C4×C53—44no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—82no—
115C2×C4×C53—84no—
116C2×C4×C53—84no—
117C2×C4×C53—84no—
118C2×C4×C53#9912yes—
119C2×C4×C53C2×C4⋊C411yescharacteristic
120C2×C4×C53#11711yescharacteristic
121#12800C5311yesmaximal, characteristic, direct product factor, Sylow
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—22no—
125C22×C53—22no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—82no—
131C22×C53#38011yescharacteristic
132C22×C53C2×#9911yescharacteristic
133C22×C53#41011yescharacteristic
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—22no—
137C4×C53—42no—
138C4×C53—42no—
139C4×C53—42no—
140C4×C53—41no—
141C4×C53—81no—
142C4×C53—81no—
143C4×C53—164no—
144#2326C2×C5311yescharacteristic
145C2×#354C2×C5312yes—
146#1799C2×C5312yes—
147#1393C2×C5312yes—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—42no—
151C2×C53—42no—
152C2×C53#179911yescharacteristic
153C2×C53#212711yescharacteristic
154C2×C53#212911yescharacteristic
155C2×#124C22×C5311yescharacteristic
156C2×C4×Q8C4×C5311yescharacteristic, complemented
157C2×C4×D4C22×C5311yescharacteristic
158C2×#99C22×C5314yes—
159#391C4×C5311yescharacteristic, complemented
160#390C22×C5311yescharacteristic
161#388—22no—
162#385—22no—
163#384C4×C5311yescharacteristic, complemented
164#380—22no—
165#376C4×C5311yescharacteristic, complemented
166#354—24no—
167#342—22no—
168C53#1280011yescharacteristic, direct product factor, Sylow
169C22×Q8C2×C4×C5311yescharacteristic
170C22×D4C23×C5311yescharacteristic
171C23×C4D4×C5312yes—
172#124—22no—
173#124D4×C5312yes—
174#124Q8×C5312yes—
175C4×Q8—42no—
176C4×D4—42no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4C2×C4×C5311yescharacteristic
181C2×C4⋊C4C2×C4×C5311yescharacteristic
182C2×C22⋊C4—22no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4D4×C5312yes—
186C2×C42—21no—
187C2×C42C2×C4×C5311yescharacteristic
188C2×C42C2×C4×C5311yescharacteristic
189C2×C42C2×C4×C5311yescharacteristic
190#99—24no—
191#99—24no—
192#95—21no—
193#95—21no—
194#95—22no—
195#95—42no—
196C24C2×D4×C5312yes—
197C2×Q8—21no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8C4⋊C4×C5312yescomplemented
201C2×D4—41no—
202C2×D4—41no—
203C2×D4C2×D4×C5311yescharacteristic
204C2×D4C2×Q8×C5311yescharacteristic
205C2×D4#41×C5312yes—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—44no—
213C22×C4—82no—
214C22×C4C22⋊C4×C5311yescharacteristic
215C22×C4C22⋊C4×C5312yes—
216C22×C4C22⋊C4×C5311yescharacteristic
217C22×C4C22×C4×C5311yescharacteristic
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—84no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—42no—
226C22⋊C4—44no—
227C22⋊C4—44no—
228C22⋊C4—82no—
229C42—22no—
230C42—21no—
231C42—41no—
232C42—42no—
233C42C4⋊C4×C5312yescomplemented
234C23—22no—
235C23—22no—
236C23—42no—
237C23—42no—
238C23C2×C22⋊C4×C5311yescharacteristic
239C23#121×C5312yes—
240C23C22⋊Q8×C5312yes—
241Q8—42no—
242Q8—42no—
243D4—41no—
244D4—41no—
245D4—42no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—82no—
266C2×C4—84no—
267C2×C4—84no—
268C2×C4—84no—
269C2×C4#99×C5312yes—
270C2×C4C2×C4⋊C4×C5311yescharacteristic
271C2×C4#117×C5311yescharacteristic
272C22—21no—
273C22—21no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—82no—
281C22C53×#38011yescharacteristic
282C22C2×#99×C5311yescharacteristic
283C22C53×#41011yescharacteristic
284C4—21no—
285C4—21no—
286C4—22no—
287C4—42no—
288C4—42no—
289C4—42no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—164no—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—42no—
298C2C53×#179911yescharacteristic
299C2C53×#212711yescharacteristic
300C2C53×#212911yescharacteristic
subgroupG/H
center, socleC22×C53C2×#99
commutatorC22×C4C22×C4×C53
Frattini, cosocleC22×D4C23×C53
prime radicalC22×D4×C53C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×C4×D4×C53 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. C2×C4×Q8×C53C4semidirect
9. C53×#391C4semidirect
13. C53×#384C4semidirect
15. C53×#376C4semidirect
49. C2×Q8×C53C4⋊C4semidirect
82. C42×C53C4⋊C4semidirect
121. #12800C53direct (N × H)
156. C2×C4×Q8C4×C53semidirect
159. #391C4×C53semidirect
163. #384C4×C53semidirect
165. #376C4×C53semidirect
168. C53#12800direct (N × H)
200. C2×Q8C4⋊C4×C53semidirect
233. C42C4⋊C4×C53semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C53×#23262
C2×C53×#3542
C53×#17992
C53×#13932
#1280053